Economics and Statistics

  Let X and Y be column vectors and the sample has the size n: The vector beta contains both the coefficient of X as well as the coefficient for the intercept. This is equivalent to writing this equation: We will assume that the expected value of the error term is 0 (this can be done by construction), and that X is uncorrelated to epsilon. Assume the following is for population data. Let's prove that statement: Since we know that the covariance between X and epsilon is 0, it follows that the expected value of X times the error term must also be 0. Since we do not have access to the actual expected value of the distribution, let's use the sample data instead: The hat on the covariance signifies that this is a MM estimator. Let's use the fact that the sample mean of epsilon is also equal to 0, then:   Using the estimation for the covariance and that the expected value of the error term equals 0: Then we have: To get this result, you must use the properties of the summation op...